I know, I know, another annoying question about calculations... 

Last time we talked about this, I mentioned this equation that was devised:
ILamp = (PF * √(VOC^2 - VLamp^2)) / ZBal
Where ILamp is RMS lamp current, PF is distortion power factor, VOC is supply voltage, VLamp is RMS lamp voltage, and ZBal is ballast impedance. I use this equation to determine the power factor:
PF = PLamp / (VLamp * ILamp)
Where PLamp is wattage. This is the definition of power factor, is it not? I am pretty sure it is. But something doesn't quite check out here. Here is an example, I am using 35W S68 North American HPS, but this same phenomena happens to most other lamps:
PF = 35 / (55 * .83)
PF = 0.77
Cool. So the distortion power factor is 0.77. So let's plug it in and see if we can get the right lamp current with all the other specs being known:
ILamp = (0.77 * √(120^2 - 55^2)) / 116.5
ILamp = .7A
So the current is .7A. But no, that is false, it is supposed to be .83A, that is an error of over 15%. I think the power factor is the problem. A while back I calculated average power factors for different technologies (hps, mv, etc) using the more complex of the two equations (not the definition of power factor, the other one). Let's try my calculated HPS average power factor of 0.913. Keep in mind that this value is averaged from across all HPS lamps of which I have specs.
ILamp = (0.913 * √(120^2 - 55^2)) / 116.5
ILamp = .84A
Well I'll be darned! That new magical PF value gave a result that is within 2% of the actual value. Very nice!

So power factor is the problem. Either the way I am calculating PF is wrong, or the way I am using PF in the big equation is wrong. Which one is it? And thank you all for your continued mathematical help, I think we are on to something here.

Check your equation. Decreasing the power factor must increase the current, ie. when calculating the current you must divide by the power factor, not multiply by it



Also, the formula with Zbal is a bit abridged, but that has smaller effect on the precision of the outcome

Z = sqrt( X^2 + R^2 ), where X is the inductive impedance, R is the resistance representing the losses (most of it is plain DC resistance of the winding wire, but there is some more to account for the core)

So the ballast itself does have its own angle between Z and X, which depends on the ballast efficiency, and does not have anything to do with the lamp PF or the complete circuit PF

When calculating, the Vdrop of the ballast consists of Vdrop(X) and Vdrop(R), which together form Vdrop(Z). However, Vdrop(R) is the one which is mostly in phase with the Vdrop of the lamp arc

So, the complete Vdrop sum would be something like :

Vline = sqrt( Vdrop(X)^2 + ( Vdrop(R) + Varc )^2 )

@Ash
Thanks for the comment!

So the PF should be in the denominator is what you are saying? Let's try it out with the 35W HPS again with the calculated PF for this particular lamp (.77):
ILamp = √(VOC^2 - VLamp^2) / (ZBal * PF)
ILamp = √(120^2 - 55^2) / (116.5 * .77)
ILamp = 1.19
Well that definitely increased the current, but way too much. That is more than double our previous 15% error. Obviously I am not doing this right lol, do I have to do something with squaring/square rooting in the denominator?

I did do some research on the inductive vs resistive impedance of the ballast, but incorporating that into the equation would probably require measuring the inductive impedance and resistive impedance of each ballast you are doing that for. I have no idea how to measure the leakage inductance of a coil while excluding the non-leakage inductance non-destructively, so maybe leaving that out would be easier. I'll consider adding that if the results we get at the end aren't close enough for our purposes.

If by PF here you mean the lamp discharge PF, 0.77 looks too low. 0.85..0.9 is normal. Could it have been calculated before with some error there ?

If you mean PF of the entire circuit, that would not appear in the equation at all (the sqrt( voltages ) in the formula already takes care of it). And 0.77 is way too high for a discharge lamp without capacitor

Also, by Zbal do you actually mean Zbal or Xbal, and how did you find it ?

@Ash
Yes, by PF I mean the distortion power factor of the lamp discharge itself, not any inductive/capacitive displacement power factor of the ballast and the lamp together.

By calculating the lamp current times the lamp voltage (.83 * 55), you get 45.65 VA. 35 W divided by 45.65 VA gives you a power factor of .77 (if I am doing this right).

By ZBal, I mean the total ballast impedance in ohms (including both inductance and DC resistance). I found it on this 2009 ANSI standard sheet, so I can only assume that it is correct and not causing my problem, though I could be wrong.

@Ash

The actual lamp power may be different from the rating. See if you can find the real lamp power, or if it lines up better if you assume something like 39W instead of 35W for the lamp. (This still have to be verified, but lets see if this would explain anything)

The formula used is correct for Xbal, not for Zbal. They are not too much far away from each other, this alone would not likely cause as high differences as you see. Unless you have a formula that can account for Xbal and Rbal separately, use Xbal and disregard Rbal

On page 44 there is a ballast curve for this lamp, and it does use wattage as the y-axis. Maybe this is just relative wattage, but at 52V is does say 35W.

Is there a way to calculate the actual usable power factor without having every single spec of this lamp?

I was thinking the same thing with the X and R. I could make a formula that uses both, but I don't know how to get the X and R of a ballast when all that is given to me in the standards is Z. So whatever, Z probably good enough.

The lamp power factor can be calculated by the same formula you used in reverse, ie when you know I and find out PF. It consists of a little displacement and a lot of distortion, so cannot be calculated just from phase angles like a purely displacement PF

It can be numerically measured if you have both Ilamp and Vlamp on a scope or power analyzer with some sampling and processing

HID ballasts are fairly efficient, so R is not significant compared to X. However, Vdrop(R) is almost directly in phase with Varc. To include it, change "Varc" to "Varc + Ilamp * Rbal" in the right places

Something im thinking about now..

In my recent visit to Dor - https://www.lighting-gallery.net/gallery/displayimage.php?album=9118&pos=0&pid=264621

We lit (for the first time since i found it) the M4A3 as is with the lamp and gear that came in it

Lamp is IIRC Philips Belgium SON-T Master 250W. Ballast is ELT Spain standard 250W ballast. Ignitor is ELT ballast dependent ignitor (connected to the ballast tap, not adding resistance in series with the lamp in run mode)

The nominal current for it is 3.0A

In our run up (measured with a fairly basic Uni-T AC+DC clamp meter, hall effect based), the current went from 3.15A for starting (seems legit, tho i expected more) to under 2.6A (quite a bit below the 3A)

Line voltage was around 220V, gear for 230V

@Ash
Yes of course, absolutely I can use the big equation with ballast impedance to determine PF, that is what I did to determine the average PF values across technologies. These averages are good enough to get ballast characteristics for most more common lamps.

But for other rarer lamps (short arc LPS, linear neon, super high pressure lamps), these averaged technology values are probably far from what the particular lamp actually is. I am sure it would not be wise of me to assume that my NA-1 has the same PF as an average SOX or SOX-E lamp, even though both are under the LPS category they surely are very different. And I am sure linear neon has a PF that is different from fluorescent even though they are physically similar. I don't have ballast specs for these less common types of lamps.

So naturally the need arises to get the power factor of a lamp which you don't have the ballast specs for, so you can use that to get the ballast specs. Let's use a 160W NE/H for example. I know that a 160W NE/H runs at 157V at 1.33A, but cannot find ballast specs for it. I normally should be able to calculate PF with just power, voltage, and current (PF=W/VA), but we have just proven that for some mysterious reason that method doesn't work accurately for discharge lamps. Dead end.

So I can't get PF because I don't have ballast characteristics, and I can't get ballast characteristics because I don't have PF. Is this a standstill? Is there anything we can do for situations like the NA-1 or NE/H to get either PF or ballast characteristics from jus tthe running specifications so we can solve the puzzle?